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Kinetic & Related Models

Open Access Articles

A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations
Alexander Alekseenko, Truong Nguyen and Aihua Wood
2018, 11(5): 1211-1234 doi: 10.3934/krm.2018047 +[Abstract](5181) +[HTML](1358) +[PDF](2466.17KB)

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision integral with \begin{document}$O(MN)$\end{document} operations, where \begin{document}$N$\end{document} is the number of the discrete velocity points and \begin{document}$M <N$\end{document}. At the base of the algorithm are nodal-discontinuous Galerkin discretizations of the collision operator on uniform grids and a bilinear convolution form of the Galerkin projection of the collision operator. Efficiency of the algorithm is achieved by applying singular value decomposition compression of the discrete collision kernel and by approximating the kinetic solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. Accuracy of the method is established on solutions to the problem of spatially homogeneous relaxation.

Kazuo Aoki, Pierre Degond and Tong Yang
2018, 11(4): i-i doi: 10.3934/krm.201804i +[Abstract](2357) +[HTML](216) +[PDF](78.89KB)
Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium
Victor A. Kovtunenko and Anna V. Zubkova
2018, 11(1): 119-135 doi: 10.3934/krm.2018007 +[Abstract](5206) +[HTML](1106) +[PDF](452.3KB)

In this paper a mathematical model generalizing Poisson-Nernst-Planck system is considered. The generalized model presents electrokinetics of species in a two-phase medium consisted of solid particles and a pore space. The governing relations describe cross-diffusion of the charged species together with the overall electrostatic potential. At the interface between the pore and the solid phases nonlinear electro-chemical reactions are taken into account provided by jumps of field variables. The main advantage of the generalized model is that the total mass balance is kept within our setting. As the result of the variational approach, well-posedness properties of a discontinuous solution of the problem are demonstrated and supported by the energy and entropy estimates.

Numerical study of a particle method for gradient flows
José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini and Gershon Wolansky
2017, 10(3): 613-641 doi: 10.3934/krm.2017025 +[Abstract](3689) +[HTML](786) +[PDF](728.3KB)

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

Self-organized hydrodynamics with density-dependent velocity
Pierre Degond, Silke Henkes and Hui Yu
2017, 10(1): 193-213 doi: 10.3934/krm.2017008 +[Abstract](3352) +[HTML](1179) +[PDF](651.5KB)

Motivated by recent experimental and computational results that show a motility-induced clustering transition in self-propelled particle systems, we study an individual model and its corresponding Self-Organized Hydrodynamic model for collective behaviour that incorporates a density-dependent velocity, as well as inter-particle alignment. The modal analysis of the hydrodynamic model elucidates the relationship between the stability of the equilibria and the changing velocity, and the formation of clusters. We find, in agreement with earlier results for non-aligning particles, that the key criterion for stability is \begin{document} $(ρ v(ρ))'≥q 0$ \end{document}, i.e. a nondecreasing mass flux \begin{document} $ρ v(ρ)$ \end{document} with respect to the density. Numerical simulation for both the individual and hydrodynamic models with a velocity function inspired by experiment demonstrates the validity of the theoretical results.

Anton Arnold, Paola Pietra and Giuseppe Toscani
2017, 10(1): i-ii doi: 10.3934/krm.201701i +[Abstract](1699) +[HTML](1267) +[PDF](1249.2KB)
A note from the Editorial Board
Editorial Board
2014, 7(1): i-i doi: 10.3934/krm.2014.7.1i +[Abstract](2203) +[PDF](77.3KB)
Robert Glassey (Bob) has decided to withdraw from the Editorial Board of KRM. Bob has been a pioneer of mathematical kinetic theory in the 80's and one of its leading figures since then. His seminal papers on the relativistic Vlasov-Poisson and Vlasov-Maxwell systems and their asymptotic stability have been a source of inspiration for many of us. His book, `The Cauchy problem in kinetic theory' has become a must for all young researchers entering the field. Bob has been involved in the Editorial Board of KRM since the beginning of the journal and has contributed to the edition of many papers. On behalf of the whole editorial board, we express our deep gratitude to him for having joined us in this adventure and contributed to the success of the journal.
Kazuo Aoki, Pierre Degond and Tong Yang
2013, 6(4): i-ii doi: 10.3934/krm.2013.6.4i +[Abstract](1523) +[PDF](504.2KB)
In 2012, we are saddened by the loss of our friend and a great mathematician, Seiji Ukai, who passed away in November. Seiji was a leading expert in the area of kinetic theory and fluid mechanics, and was always kind and humble so that he was beloved by his colleagues, collaborators and researchers in these research areas. As Cédric Villani said in his article, ``Good-bye my friend...(In Memoriam Seiji Ukai)", ``He was a great mathematician and a beautiful soul".

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Seiji Ukai
Yan Guo
2013, 6(4): iii-iii doi: 10.3934/krm.2013.6.4iii +[Abstract](1573) +[PDF](80.8KB)
Seiji Ukai has made fundamental contributions to the mathematical study for the Boltzmann equation. His seminal construction of global solutions near Maxwellian opens the era for the study of non-homogeneous Boltzmann solutions, his work on a gas passing an obstacle remains a penetrating classical result, and his study of inverse power laws without angular cutoff marks a breakthrough in the field.
    I first met Seiji in 1995 at Oberwolfach. I remember that he smoked a lot and we had to stand in the cold for discussions. He was gentle and soft-spoken, paused and thought carefully before answering my questions. We met several years later again at Oberwolfach, when I presented my construction of global smooth solutions to the compressible Euler-Poisson system. He was excited and relaxed, and with Bob Glassey, we talked a lot about problems in fluids.
    Motivated by my desire to understand collision effects in a plasma, I decided to learn and work on the Boltzmann equation. I was greatly influenced by his elegant survey paper `Solutions of the Boltzmann Equation', and later found a different way to control the macroscopic part of Boltzmann solutions. I was very encouraged by the positive feedbacks from the Boltzmann community, in particular, Desvillettes, Illner, Ukai and Villani, among others.
    In 2001, I visited Japan, passing through Tokyo. Seiji insisted meeting with me at my hotel in Tokyo (he had to travel for two hours for that!). He took me out for dinner and we chatted a lot about everything. As with many of my Japanese friends, when conversations go deeper, sometimes we resort to Chinese characters for better communications. His knowledge in Chinese characters was so impressive, and I remember clearly our discussion about the Chinese character `head'. He was shocked to learn its simplified form currently in use, which is so much different from its traditional counterpart. Shooking his head in disbelief, he wanted to know the exact reason behind such a simplification. A true scholar, Seiji was so meticulous about every detail! It got quite late and I stood outside my hotel, watching him walking quickly towards the subway station.
    I have not seen Seiji too much in recent years. I am always grateful for his encouragements and mathematical insights, and will always remember the lovely evening we spent together in 2001.
On the dynamics of social conflicts: Looking for the black swan
Nicola Bellomo, Miguel A. Herrero and Andrea Tosin
2013, 6(3): 459-479 doi: 10.3934/krm.2013.6.459 +[Abstract](3183) +[PDF](702.8KB)
This paper deals with the modeling of social competition, possibly resulting in the onset of extreme conflicts. More precisely, we discuss models describing the interplay between individual competition for wealth distribution that, when coupled with political stances coming from support or opposition to a Government, may give rise to strongly self-enhanced effects. The latter may be thought of as the early stages of massive unpredictable events known as Black Swans, although no analysis of any fully-developed Black Swan is provided here. Our approach makes use of the framework of the kinetic theory for active particles, where nonlinear interactions among subjects are modeled according to game-theoretical principles.
A note from the Editorial Board
Editorial Board
2012, 5(1): i-i doi: 10.3934/krm.2012.5.1i +[Abstract](2223) +[PDF](21.3KB)
From this year, Professor Seiji Ukai will step down from the managing editorial board of KRM. Seiji Ukai is a world leading expert in the field of mathematical theories of kinetic equations. He is the first one who proved the global existence of solutions to the space inhomogeneous Boltzmann equation in 1974, and he has been making important contributions to this area, including his series of recent works on the Boltzmann equation without angular cutoff. We are deeply indebted to the contributions that Seiji made to the area and the journal and wish him all the best in the future.

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Paola Pietra, Eric Polizzi, Fabrice Deluzet, Jihène Kéfi, Olivier Pinaud, Claudia Negulescu, Nicolas Vauchelet, Raymond El Hajj, Clément Jourdana and Stefan Possanner
2011, 4(4): i-iii doi: 10.3934/krm.2011.4.4i +[Abstract](1486) +[PDF](892.5KB)
On July 5-th 2010, Naoufel Ben Abdallah tragically passed away at the age of 41. He was an extremely talented mathematician with a deep interest in applications, and an uncommon ability in creating links for interdisciplinary research.

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Maria Lampis and Mario Pulvirenti
2011, 4(1): i-v doi: 10.3934/krm.2011.4.1i +[Abstract](1649) +[PDF](2567.3KB)
On January 7-th, 2010, Carlo Cercignani, Professor of Mathematical Physics at Politecnico di Milano, passed away in Milan, after a long illness.
   If one ought to indicate a single man as a reference point for the development of kinetic theory in the last 50 years from a mathematical, physical, historical and, more generally, cultural point of view, there is no doubt that everybody would think of Carlo Cercignani. We have not only lost a very reputed and loved colleague, but also a pivotal figure for all of us working in the field.

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José Antonio Carrillo, Peter Markowich and Lorenzo Pareschi
2010, 3(1): i-i doi: 10.3934/krm.2010.3.1i +[Abstract](1571) +[PDF](27.1KB)
The idea of putting together a special issue of KRM in honor of Giuseppe Toscani's 60$th$ birthday was received with great enthusiasm by the "kinetic" community and by Giuseppe's scientific colleagues and friends from other mathematical communities. There is a common feeling throughout that we have learned a great deal from the ideas, the methodologies and the scientific conclusions of Giuseppe Toscani's work.
   The papers related to Toscani's research in this special issue are authored by collaborators, friends and students of Giuseppe. The topics include classical and non classical applications of kinetic theory, entropy-entropy dissipation methods, asymptotic techniques and numerical simulations.The articles in this volume are ordered alphabetically by names of authors.

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Existence and sharp localization in velocity of small-amplitude Boltzmann shocks
Guy Métivier and K. Zumbrun
2009, 2(4): 667-705 doi: 10.3934/krm.2009.2.667 +[Abstract](1874) +[PDF](465.9KB)
Using a weighted $H^s$-contraction mapping argument based on the macro-micro decomposition of Liu and Yu, we give an elementary proof of existence, with sharp rates of decay and distance from the Chapman-Enskog approximation, of small-amplitude shock profiles of the Boltzmann equation with hard-sphere potential, recovering and slightly sharpening results obtained by Caflisch and Nicolaenko using different techniques. A key technical point in both analyses is that the linearized collision operator $L$ is negative definite on its range, not only in the standard square-root Maxwellian weighted norm for which it is self-adjoint, but also in norms with nearby weights. Exploring this issue further, we show that $L$ is negative definite on its range in a much wider class of norms including norms with weights asymptotic nearly to a full Maxwellian rather than its square root. This yields sharp localization in velocity at near-Maxwellian rate, rather than the square-root rate obtained in previous analyses.
Preface I
Claude Bardos, Francois Golse and Paul Martin
2009, 2(1): i-iv doi: 10.3934/krm.2009.2.1i +[Abstract](1324) +[PDF](51.9KB)
Basil Nicolaenko was born in Paris in 1942, to Russian immigrants. A former Cossack, his father first took a blue collar job at the nearby Renault car factory, and later became a taxi driver. Basil grew up in his parents' studio, in the southwest of Paris, where he lived until the age of 20. His high school education was supported by a scholarship of the French Government. However, things were far from simple: because of their still relatively poor command of French, his parents could hardly help him with the subtleties of the French bureaucracy. Thus, at the age of 11, Basil found himself having to argue in person with the uncooperative headmaster of Lycée Buffon to gain admission to 6th grade -- a fact he still remembered with unconcealed indignation more than 50 years later.

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Preface II
Pierangelo Marcati
2009, 2(1): v-vii doi: 10.3934/krm.2009.2.1v +[Abstract](1488) +[PDF](46.0KB)
Angelo Marcello Anile was an internationally known applied mathematician and mathematical physicist, passed away on November 16, 2007 after one very difficult year. Angelo Marcello Anile was an internationally known applied mathematician and mathematical physicist, passed away on November 16, 2007 after one very difficult year.
I personally had the privilege to know Marcello long time ago, through our common friend and colleague Giovanni Russo, who was a former student of Marcello freshly hired that time from my Department.
I was impressed from his results on hydrodynamical models for Semiconductor Devices and, together Peter Markowich and Roberto Natalini, on December 1993 we invited him at the CNR IAC Institute Mauro Picone, where we were organizing a workshop on these topics. Marcello gave a beautiful minicourse regarding An extended thermodynamic framework for the hydrodynamical modeling of semiconductors.

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Pierre Degond, Seiji Ukai and Tong Yang
2008, 1(1): i-ii doi: 10.3934/krm.2008.1.1i +[Abstract](1872) +[PDF](26.2KB)
Kinetic theory is probably one of the most efficient and important theories allowing to bridge the microscopic and macroscopic descriptions of a variety of dynamical phenomena in many fields of science, technology, and more generally, in virtually all domains of knowledge. Originally rooted in the theory of rarefied gases since the seminal works of Boltzmann and Maxwell in the 19th century, followed by landmarks established by Hilbert, Chapman and Enskog, Carleman, Grad, and more recent mathematicians, kinetic theory has expanded to many new areas of applications, ranging from physics to economics and social sciences including especially modern fields such as biology, epidemiology, and genetics.

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